Olympiad problems

2023 Bulgaria EGMO TST, 4

Each two-digit is number is coloured in one of $k$ colours. What is the minimum value of $k$ such that, regardless of the colouring, there are three numbers $a$, $b$ and $c$ with different colours with $a$ and $b$ having the same units digit (second digit) and $b$ and $c$ having the same tens digit (first digit)?

2025 Israel National Olympiad (Gillis), P3

Tags: number theory , Perfect Squares , Digits

Bart wrote the digit "$1$" $2024$ times in a row. Then, Lisa wrote an additional $2024$ digits to the right of the digits Bart wrote, such that the resulting number is a square of an integer. Find all possibilities for the digits Lisa wrote.

2013 Czech And Slovak Olympiad IIIA, 4

Tags: combinatorics , game , Digits , Digit

On the board is written in decimal the integer positive number $N$. If it is not a single digit number, wipe its last digit $c$ and replace the number $m$ that remains on the board with a number $m -3c$. (For example, if $N = 1,204$ on the board, $120 - 3 \cdot 4 = 108$.) Find all the natural numbers $N$, by repeating the adjustment described eventually we get the number $0$.

2009 Regional Olympiad of Mexico Center Zone, 4

Tags: Perfect Square , Perfect Squares , consecutive , Digits , number theory

Let $N = 2 \: \: \underbrace {99… 9} _{n \,\,\text {times}} \: \: 82 \: \: \underbrace {00… 0} _{n \,\, \text {times} } \: \: 29$. Prove that $N$ can be written as the sum of the squares of $3$ consecutive natural numbers.

1986 China Team Selection Test, 3

Tags: algebra , Number theoretic functions , China , TST , decimal representation , Digits

Given a positive integer $A$ written in decimal expansion: $(a_{n},a_{n-1}, \ldots, a_{0})$ and let $f(A)$ denote $\sum^{n}_{k=0} 2^{n-k}\cdot a_k$. Define $A_1=f(A), A_2=f(A_1)$. Prove that: [b]I.[/b] There exists positive integer $k$ for which $A_{k+1}=A_k$. [b]II.[/b] Find such $A_k$ for $19^{86}.$

1994 Bundeswettbewerb Mathematik, 1

Tags: pigeonhole , decimal representation , real number , Digits , algebra

Given eleven real numbers, prove that there exist two of them such that their decimal representations agree infinitely often.

2015 India PRMO, 15

Tags: number theory , Digits , prime numbers

$15.$ Let $n$ be the largest integer that is the product of exactly $3$ distinct prime numbers, $x,y,$ and $10x+y,$ where $x$ and $y$ are digits. What is the sum of digits of $n ?$

2024 Bundeswettbewerb Mathematik, 2

Tags: number theory , number theory proposed , Digits , Perfect Squares , Perfect Square

Can a number of the form $44\dots 41$, with an odd number of decimal digits $4$ followed by a digit $1$, be a perfect square?

2013 NZMOC Camp Selection Problems, 7

Tags: number theory , Digits

In a sequence of positive integers an inversion is a pair of positions such that the element in the position to the left is greater than the element in the position to the right. For instance the sequence $2,5,3,1,3$ has five inversions - between the first and fourth positions, the second and all later positions, and between the third and fourth positions. What is the largest possible number of inversions in a sequence of positive integers whose sum is $2014$?

2015 Saudi Arabia JBMO TST, 2

Tags: combinatorics , Digits , odd

Let $A$ and $B$ be the number of odd positive integers $n<1000$ for which the number formed by the last three digits of $n^{2015}$ is greater and smaller than $n$, respectively. Prove that $A=B$.

2024 Dutch IMO TST, 4

Tags: number theory , number theory proposed , blackboard , Digits

Initially, a positive integer $N$ is written on a blackboard. We repeatedly replace the number according to the following rules: 1) replace the number by a positive multiple of itself 2) replace the number by a number with the same digits in a different order. (The new number is allowed to have leading digits, which are then deleted.) [i]A possible sequence of moves is given by $5 \to 20 \to 140 \to 041=41$.[/i] Determine for which values of $N$ it is possible to obtain $1$ after a finite number of such moves.

1962 Dutch Mathematical Olympiad, 4

Tags: floor function , number theory , Digits , last digits

Write using with the floor function: the last, the second last, and the first digit of the number $n$ written in the decimal system.

2019 Cono Sur Olympiad, 2

Tags: number theory , Digits

We say that a positive integer $M$ with $2n$ digits is [i]hypersquared[/i] if the following three conditions are met: [list] [*]$M$ is a perfect square. [*]The number formed by the first $n$ digits of $M$ is a perfect square. [*]The number formed by the last $n$ digits of $M$ is a perfect square and has exactly $n$ digits (its first digit is not zero). [/list] Find a hypersquared number with $2000$ digits.

2007 Cuba MO, 5

Tags: number theory , Digits , divides , power of 2

Prove that there is a unique positive integer formed only by the digits $2$ and $5$, which has $ 2007$ digits and is divisible by $2^{2007}$.

1980 IMO, 3

Tags: modular arithmetic , combinatorics , decimal representation , Digits , IMO Shortlist

Find the digits left and right of the decimal point in the decimal form of the number \[ (\sqrt{2} + \sqrt{3})^{1980}. \]

1968 Bulgaria National Olympiad, Problem 3

Tags: number theory , Digits , Bases

Prove that a binomial coefficient $\binom nk$ is odd if and only if all digits $1$ of $k$, when $k$ is written in binary, are on the same positions when $n$ is written in binary. [i]I. Dimovski[/i]

2022 Denmark MO - Mohr Contest, 2

Tags: number theory , palindrome , Digits

A positive integer is a [i]palindrome [/i] if it is written identically forwards and backwards. For example, $285582$ is a palindrome. A six digit number $ABCDEF$, where $A, B, C, D, E, F$ are digits, is called [i]cozy [/i] if $AB$ divides $CD$ and $CD$ divides $EF$. For example, $164896$ is cozy. Determine all cozy palindromes.

1983 Bundeswettbewerb Mathematik, 3

Tags: number theory , Digits

A real number is called [i]triplex[/i] if it has a decimal representation in which none of $0$ and $3$ different digit occurs. Prove that every positive real number is the sum of nine triplex numbers.

2015 Belarus Team Selection Test, 2

Tags: Sequence , number theory , Last digit , Digits

In the sequence of digits $2,0,2,9,3,...$ any digit it equal to the last digit in the decimal representation of the sum of four previous digits. Do the four numbers $2,0,1,5$ in that order occur in the sequence? Folklore

2013 Chile National Olympiad, 1

Tags: combinatorics , Digits , number theory

Find the sum of all $5$-digit positive integers that they have only the digits $1, 2$, and $5$, none repeated more than three consecutive times.

2019 Gulf Math Olympiad, 2

Tags: number theory , Digits , multiple

1. Find $N$, the smallest positive multiple of $45$ such that all of its digits are either $7$ or $0$. 2. Find $M$, the smallest positive multiple of $32$ such that all of its digits are either $6$ or $1$. 3. How many elements of the set $\{1,2,3,...,1441\}$ have a positive multiple such that all of its digits are either $5$ or $2$?

2023 India Regional Mathematical Olympiad, 4

Tags: Digits , number theory , combinatorics

The set $X$ of $N$ four-digit numbers formed from the digits $1,2,3,4,5,6,7,8$ satisfies the following condition: [i]for any two different digits from $1,2,3,4,,6,7,8$ there exists a number in $X$ which contains both of them. [/i]\\ Determine the smallest possible value of $N$.

2018 Estonia Team Selection Test, 6

Tags: number theory , Digits

We call a positive integer $n$ whose all digits are distinct [i]bright[/i], if either $n$ is a one-digit number or there exists a divisor of $n$ which can be obtained by omitting one digit of $n$ and which is bright itself. Find the largest bright positive integer. (We assume that numbers do not start with zero.)

1989 Swedish Mathematical Competition, 1

Tags: number theory , Digits , Perfect Square

Let $n$ be a positive integer. Prove that the numbers $n^2(n^2 + 2)^2$ and $n^4(n^2 + 2)^2$ are written in base $n^2 +1$ with the same digits but in opposite order.

1981 Bundeswettbewerb Mathematik, 1

Tags: Digits , number theory , powers

Let $a$ and $n$ be positive integers and $s = a + a^2 + \cdots + a^n$. Prove that the last digit of $s$ is $1$ if and only if the last digits of $a$ and $n$ are both equal to $1$.